Mathematical analysis : complements

Teacher(s)

Absil Pierre-Antoine; Van Schaftingen Jean;

Language

French

Main themes

This course covers themes in mathematical analysis (measure theory, functional analysis and function spaces) that play a role in the foundations of various areas of applied mathematics such as dynamical systems, partial differential equations, optimal control, scientific computing, stochastic processes and financial mathematics.

Aims

At the end of this learning unit, the student is able to :
1	AA 1.1, 1.2, 1.3, 3.1. At the end of the course, the student will be able to: 1. by means of examples, statements and mathematical proofs, describe infinite-dimensional spaces, including their operators and convergence notions, and compare them to finite dimensional spaces, 2. apply definitions and results of measure theory to the study of function spaces and probability theory, 3. use advanced concepts of measure theory and functional analysis in applied mathematics.

Content

Important concepts and results within the main themes of the course,
such as:

• Measure theory, Lebesgue integral, convergence theorems,
• Complete metric spaces, Banach spaces and Hilbert spaces, spaces of continuous functions, spaces of integrable functions,

• Continuous linear mappings, weak convergence, Riesz representation theorem, notions of spectral theory,
• Distributions and Sobolev spaces.

Teaching methods

Due to the COVID-19 crisis, the information in this section is particularly likely to change.

The course includes interactive lectures and exercises. The emphasis is
on critical understanding of the theory and active problem solving.

Evaluation methods

Due to the COVID-19 crisis, the information in this section is particularly likely to change.

• Homeworks, exercises, tests or practical work carried out during the semester
• Exam

More elaborate information on the evaluation procedure is given in the
course outline, made available on Moodle at the beginning of the
academic year.

Other information

Bibliography

Livre de référence : Gerald Teschl, "Topics in Real and Functional Analysis" disponible gratuitement en ligne à l'adresse
(https://www.mat.univie.ac.at/~gerald/ftp/book-fa/).

Faculty or entity

MAP